Planar QSOs

We begin with the simpler case of planar QSOs before extending the study to three-dimensional orbits.

x0Vs.y˙0

Let us begin with the stability analysis of planar QSOs with the variation ofx0andy˙0. This analysis is performed for the case of initial true anomalyf0 = 0, i.e., Phobos’ perigee passage. The other initial parametersy0,z0,x˙0,z˙0are set to zero.

The orbital amplitude in the x-y plane, α, is best measured in the passage of the orbit byy0 = 0 because experimentation suggests that the displacement of the QSO in thexdirection is much smaller than in they direction, i.e.,δ_x << δ_y. The analysis of the QSO stability with the distance to the moon on thex-axis provides a much better estimation of the value of the orbital amplitudeαthan the same analysis when carried out in they-axis.

The results presented in figure 4.4 suggest an almost linear relation betweenx₀andy˙₀for sufficiently stable QSOs. These FLI maps also suggest that amplitudes as large as 400 km are possible although the stability iny˙0seems to decrease with the increase of the amplitudeα≈x0.

The zoom-in of the FLI map in 4.4(b) suggests that the minimum amplitude for sufficiently stable quasi-synchronous orbits is in the interval 35–40 km, a value not far from the value of 29.43 km that we estimated in the analytical approach of our problem.

y₀Vs. x˙₀

We now analyze the stability of QSOs with the variation ofy₀andx˙₀. The initial true anomaly and the other initial parameters are set to zero. We are interested in the comparison with the previous case to

(a) Full View

(b) (Zoom-In)

Figure 4.4.FLI Maps for 2D QSOs in the Mars-Phobos system with initial conditions negativex0 and positivey˙0. The remainder of the initial parameters are set to zero.

assess which of the two cases presents as a better candidate for the transfer to a QSO or for the escape from a QSO.

The FLI map on the analyzed parametersy₀ andx˙₀(fig. 4.5) suggests that the stability region in this case is larger (regarding the analyzed parameters) than the previous case wherex₀andy˙₀were varied and, thus, the QSO stability is less sensitive to the variation of the initial velocity when maintaining the original direction. For now, no conclusions can be made regarding the sensitivity of the QSO stability to changes in the direction of the velocity.

Note that the maximum distance for which sufficiently stable QSOs are found is, roughly, 270 km, a value that is not in accordance with the value obtained in the previous analysis since, from our analytical approach, it is predicted that a QSO reaches larger distances in the y-axis than in the x-axis. The previous case found sufficiently stable at larger distance in the x-axis. A possible explanation is that large amplitude QSOs require a velocity in the y-axis to remain stable. A hypothesis that is analyzed later with the variation of this velocity.

The light-color areas in the FLI map are due to orbits that escaped or collided against Phobos. Chaotic

Figure 4.5.FLI Maps for 2D QSOs in the Mars-Phobos system with initial conditions negativey0 and negativex˙0. The remainder of the initial parameters are set to zero.

regions are known to coexist with stability regions near the libration points (Villac and Aiello, 2005) but these chaotic regions were not found with the resolution used for the computation of the FLI maps.

However, we are not interested in the study of the stability of the region near the libration points, just above Phobos’ surface, due to the restrictions of our dynamics model which neglected perturbations which are considerable at such close distances.

(a)x˙0=−20m/s (b)x˙0=−50m/s

Figure 4.6.Trajectories of two planar QSOs with entry point aty0=−100km and initial tangential velocity

˙

x0=−20m/s andx˙0=−50m/s.

From the FLI map in figure 4.5, two sample QSOs are presented in figure 4.6 with initial position y₀ =−100 km and with initial tangential velocityx˙₀ =−20m/s andx˙₀ = −50m/s, respectively. Note that the maximum distance of the QSO change with the initial velocity. The sample QSOs suggest that a larger initial velocity at the same initial distance results in a QSO with a larger amplitudeαand a larger variation of the displacement of its originδ_y.

Relation Between the Mean Motion of the QSO and Phobos

The dynamics model developed in our work does not take into account the oblateness of Mars, the solar radiation pressure, the irregular shape of Phobos or the rotation of the moon. One of the conse- quences of neglecting these perturbations is the nonexistence of resonant orbits, i.e., the orbits do not lock in the resonances 2:1, 3:2, 4:3 — ratio of the QSO’s with Phobos’ mean motion — as studied in (Wiesel, 1993). These type of QSOs are particularly interesting for their proximity to the moon.

Nevertheless, the distances at which orbits with a relation between its mean motion and the moon’s mean motion formed by two small integers could be found as discussed in section 3.3.2. We present in table 4.6 the distances found for orbits with these relations of the mean motion found analytically as well as the ones found numerically with our model. The obtained values are in accordance with some of the resonant orbits found in (Wiesel, 1993) although there is not enough data to assess if these intervals change in a more realistic model. An example of a QSO with a mean orbital motion that is twice the mean orbital motion of Phobos is presented in 4.7.

ωQSO/ωP h

Numerical Results Analytical Results x0[km] y˙0[m/s] Amplitude [km]

2:1 [−20,−17] [11.7,11.9] 17.117 3:2 [−26,−24] [13.4,13.7] 21.5661 4:3 [−31,−29] [15.0,15.3] 24.687

Table 4.6.Comparison between the prediction for the resonant orbits amplitude and the intervals where they where found by numerical exploration.

Figure 4.7.Trajectory of a QSO with a mean orbital motion that is twice the mean orbital motion of Phobos with initial conditionsy0=−22.1km andx˙0=−8.6m/s.

y0Vs. y˙0

We now study the stability of QSOs with the variation of the parametersy₀andy˙₀. This analysis is performed in Phobos’ perigee passage,f₀ = 0, withx₀,z₀, andz˙₀set to zero. The initial velocityx˙₀ is derived from a defined value of the initial value of the modified Jacobi integral. From equation (4.5), the

initial modified Jacobi integral forf0= 0is

C₀= 2 Ω⁰₀

1 +e−V₀² (4.7)

whereV0= ( ˙x²₀+ ˙y₀²+ ˙z²₀)^1/2is the spacecraft initial velocity andΩ⁰₀is the initial amended potential

Ω⁰₀= 1

2((x0+µ−1)²+y²₀−e z₀²) +1−µ r1₀

+ µ r2₀

+1

2µ(1−µ) (4.8)

withr1₀andr2₀ being the initial distance of the spacecraft to Mars and Phobos, respectively.

We compute the range ofC0 for which sufficiently stable QSOs were found in figure 4.5 which gives the interval [2.9546; 2.9554]. With these values in mind, FLI maps are now plotted for three different values ofC0:2.9547,2.9550, and2.9553. Note that smaller values ofC0imply larger velocities and, thus, larger amplitudes. The modified Jacobi integral depends on the energy of the spacecraft byC=−2E.

In the analytical approach we estimated stability conditions for QSOs depending on f0 (equations 3.29). These conditions did not need to be fulfilled strictly since they referred to the ’unperturbed initial conditions’. Recall this condition forf0= 0in dimensional coordinates

˙

y0≈ −2 +e

1 +en x0 (4.9)

For an initial conditionx₀ = 0, sufficiently stable orbits are predicted to exist only for small values of

˙

y0. No stability conditions were imposed on the other coordinates.

The FLI maps presented in figure 4.8 suggest that, for low and medium amplitude orbits, the velocity ony can not deviate much from zero. The FLI maps also suggest that by adding a positive velocity in theydirection in large amplitude QSOs it is possible to obtain sufficiently stable orbits as far as 900 km.

An example of a large amplitude QSO is presented in figure 4.9. These QSOs extend over a very large region and, thus, are good candidates for transfer QSOs to reach small amplitude QSOs.

The study of the FLI maps in figures 4.8 and 4.5 suggest that the stability condition in (4.9) only holds for initial distances up to 270 km in the y-axis which is an indication of the validity of the first order approximations performed in the analytical approach.

In conjunction with the FLI map presented in figure 4.5, the present analysis also suggests that QSOs, when in the negativeysemi-axis forf₀= 0, are more sensitive to changes in the velocity in theydirection than in thexdirection.

Furthermore, it is important to discuss the effect of the modified Jacobi integral value, C₀, which depends on the initial conditions and are a measure of the energy of the spacecraft. Figure 4.8(c) suggests that for large values of C0 (and thus smaller velocities) the maximum distance a QSO can achieve is smaller.

(a)C0= 2.9547

(b)C0= 2.9550

(c)C0= 2.9553

Figure 4.8.FLI Maps for 2D QSOs in the Mars-Phobos system with initial conditions y0 and y˙0. The initial parametersx0,z0andz˙0are set to zero,x˙0is computed from the initial modified Jacobi integralC0

(a) FLI Map (C0= 2.9547) (b) QSO (y0= 900km)

Figure 4.9.FLI Map of 2D QSOs with large amplitudes in the Mars-Phobos system with initial conditions x0= 0,C0= 2.5547and tangential and radial velocity. Sample QSO trajectory withy0= 900 km,x˙0=−55m/s, andy˙0= 15m/s

Initial True Anomaly

Up until now we have been analyzing the stability of QSOs for an initial true anomaly f0 = 0 cor- responding to Phobos’ perigee passage. However, we are interested in analyzing how the initial true anomaly influences the stability of QSOs.

We repeat the analysis of the variation of y₀ withx˙₀ for three different values of the initial anomaly:

0, π, andπ/2. The first value has already been analyzed and serves as a basis of comparison. The second value corresponds to Phobos’ apogee passage and is also a case of interest due to mission design purposes. The third value was chosen to analyze a less ordinary case.

The FLI maps in figure 4.10 suggest that the initial true anomaly plays an important role in the stability of QSOs. The stability regions (in the varying parameters y0 andx˙0) are larger for the casesf0 = 0 andf0=πwhich suggests that might be easier, from a stability point of view, to enter in a QSO during Phobos passage in the perigee or apogee. The stability region in the latter appears to be slightly larger.

(a)f0= 0

(b)f0=π

(c)f0=^π₂

Figure 4.10.FLI Maps for 2D QSOs in the Mars-Phobos system with initial conditionsy0 and x˙0. The initial parametersx0,z0,y˙0, andz˙0are set to zero. The FLI maps are presented for different values of the initial true anomalyf0